Category:Definitions/Borel Sigma-Algebras
This category contains definitions related to Borel Sigma-Algebras.
Related results can be found in Category:Borel Sigma-Algebras.
Topological Space
Let $\struct {S, \tau}$ be a topological space
The Borel sigma-algebra $\map \BB {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$.
That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.
Metric Space
Let $\struct {S, \norm {\, \cdot \,} }$ be a metric space.
The Borel sigma-algebra (or $\sigma$-algebra) on $\struct {S, \norm {\, \cdot \,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset S$.
By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on a topological space.
Pages in category "Definitions/Borel Sigma-Algebras"
The following 5 pages are in this category, out of 5 total.